Finally, we prove discreteness when \(X\)  is locally compact. By [1]}, the metric completion of \(X\)  is a \(\mathrm {CAT}(0)\)  space. Since the hypotheses of Theorem REF  are unaffected under this completion,
we may assume that \(X\)  is complete. Since \(X\)  is complete and locally compact, it is proper by the Hopf-Rinow Theorem [1]}. Under the topology of pointwise convergence, we then have that \({\rm Isom}(X)\)  is a topological group [3]} and it acts continuously on \(X\)  [3]}. Moreover, it follows from [1]} that each of the subsets \(X_i^\pm \)  is closed in \(X\) . The second part of Lemma REF  then completes the proof of the Theorem.
